Let `f:R rarr R` be a function defined as `f(x)=(x^(2)-6)/(x^(2)+2)`, then f is

Options

A. one - one but not onto
B. one - one and onto
C. onto but not one - one
D. neither one - one nor onto

Answer

To determine the properties of the function \( f(x) = \frac{x^2 - 6}{x^2 + 2} \), we will analyze whether it is one-one (injective) and onto (surjective).

### Step 1: Check if the function is one-one (injective)

To check if \( f \) is one-one, we need to see if \( f(a) = f(b) \) implies \( a = b \).

1. Calculate \( f(5) \):
\[ f(5) = \frac{5^2 - 6}{5^2 + 2} = \frac{25 - 6}{25 + 2} = \frac{19}{27} \]

2. Calculate \( f(-5) \):
\[ f(-5) = \frac{(-5)^2 - 6}{(-5)^2 + 2} = \frac{25 - 6}{25 + 2} = \frac{19}{27} \]

Since \( f(5) = f(-5) \) and \( 5 \neq -5 \), this shows that the function is not one-one.

### Step 2: Check if the function is onto (surjective)

To check if \( f \) is onto, we need to determine the range of \( f \) and compare it with its codomain, which is \( \mathbb{R} \).

1. Set \( y = f(x) \):
\[ y = \frac{x^2 - 6}{x^2 + 2} \]

2. Cross-multiply to eliminate the fraction:
\[ y(x^2 + 2) = x^2 - 6 \]
Rearranging gives:
\[ yx^2 + 2y = x^2 - 6 \implies (y - 1)x^2 = -2y - 6 \]
Thus,
\[ x^2 = \frac{-2y - 6}{y - 1} \]

3. For \( x^2 \) to be non-negative:
\[ \frac{-2y - 6}{y - 1} \geq 0 \]

4. Determine the critical points:
- The numerator \( -2y - 6 = 0 \) gives \( y = -3 \).
- The denominator \( y - 1 = 0 \) gives \( y = 1 \).

5. Analyze the sign of the expression:
- For \( y < -3 \): both numerator and denominator are negative, so the expression is positive.
- For \( -3 < y < 1 \): the numerator is negative and the denominator is positive, so the expression is negative.
- For \( y > 1 \): both numerator and denominator are negative, so the expression is positive.

6. Therefore, the range of \( f \) is \( (-\infty, -3] \cup (1, \infty) \).

Since the codomain is \( \mathbb{R} \) and the range is not equal to the codomain, \( f \) is not onto.

### Conclusion

The function \( f(x) = \frac{x^2 - 6}{x^2 + 2} \) is neither one-one nor onto.